Our New Web Site

Any instructor of computer science or mathematics is invited
to test any of our projects in the classroom for courses in
discrete mathematics, combinatorics, graph theory, algorithm design,
logic, abstract algebra, foundations of mathematics, or the history
of mathematics. Here is a
list of projects
to be written under a Phase II expansion proposal.
For further information, contact either
Jerry Lodder or Desh Ranjan above.

This site offers written curricular materials,
based on primary historical sources, for beginning and advanced
undergraduate courses in discrete mathematics and computer science.
Such courses, which often cover combinatorics, deductive reasoning
(logic) and algorithmic thought, draw a variety of majors,
ranging from computer science, mathematics,
the physical sciences and engineering to secondary education.
Traditional methods of instruction follow ``The Modern American Discrete
Mathematics Text,'' which although
thorough and mathematically precise, present the material as a
fast-paced news reel of facts and formulae, often memorized by the
students, with the text itself offering only
passing mention of the motivating problems and original work which
eventually found resolution in modern concepts such as induction,
recursion, or algorithm.

Our site offers written projects for a course in discrete or
finite mathematics with the projects containing excerpts from
primary sources for students to read along with a
sequence of directed questions which illuminate how the source develops
key mathematics ideas. Particular advantages of the historical
approach include providing context and direction for the subject
matter, honing the students' verbal and deductive skills through
reading the original work of some of the greatest minds in history,
and the rediscovery of the conceptual roots common to discrete
mathematics and computer science. Additionally, students practice the
skill of moving from verbal descriptions of problems to precise mathematical
formulations, and must often recognize an organizing concept for a
detailed procedure. Such abilities are vital not only for mathematics
and the sciences, but especially today for software engineers, who
must translate a verbal request into precise code changes, and then
realize what effect these changes will have on the global structure
of a large program or a body of interacting programs.

When working on an historical project, a student spends one to
two weeks, either as an individual or in a group, writing a detailed
paper, with two or three projects together counting for a significant
percentage (about 50%) of the course grade, and often taking the
place of two one-hour examinations. A written paper allows the
students to organize their own thoughts, react to the original sources in
much the same way as the contemporaries of the historical masterpiece,
and explore the development of ground-breaking ideas. The written
historical project builds naturally on the idea of calculus projects
[CG],
used extensively here at New Mexico State University.

To exemplify the historical approach, what better way to see induction
in action than to read from Blaise Pascal's (1623--1662) ``Treatise on
the Arithmetical Triangle'' [P1],
in which Pascal notices a
pattern in the ratio between consecutive entries in the same base of
his triangle (Pascal's Twelfth Consequence), and then claims that the
pattern holds in every base of the triangle, since if it holds in a
given base, then it persists to the next base. It would be a
wonderful exercise for students to read Pascal's original statement,
explore its truth with a few concrete numerical values, and then
grapple with the logic behind what today is known as induction. The
students (and teachers) witness first hand the genesis of abstract
concepts, and with a bit of historical background, realize that Pascal
was motivated by solving actual problems of his day, such as computing
the odds in a game of chance, or finding the summation of powers,
with the eventual goal of computing area under curves.
Another advantage to reading Pascal's original work is that
the ratios between consecutive entries that he derives lead to the modern
formula for binomial coefficients, a formula which is often announced
today without exploration of its historical origin.

By contrast, modern textbooks often give an abstract logical
formulation of induction first, and then, as homework, simply ask the
students to verify statements the (textbook) author already knows to
be true, such as the sum of i2 from i
= 1 to n is
(1/3)n3 + (1/2)n2 +
(1/6)n,
without mentioning why or how anyone discovered or
originally proved this formula.

It is being increasingly recognized that an historical point of view
can provide context, motivation and direction to mathematics courses.
For instance, an historical perspective is
being advocated in teaching calculus [K1,
K2,
R1,
R2],
while original source materials are being incorporated
in a variety of ways into calculus instruction [L1,
O], and
new texts are emphasizing the importance of studying original sources
[BG,
Ca,
F,
K3,
K4,
LP].
Also see the web page
``Teaching with Original Historical Sources in Mathematics''.
This site, however, focuses on historical projects for undergraduate
discrete mathematics courses. While there are presently excellent
accounts of the history of algorithms such as [Ch], or the
history of logic
[Da,
Gr],
such texts do not focus on the needs
of undergraduates and contain no curricular materials ready for classroom
use.

The benefits of an
historical point of view have been explained very convincingly
by Miguel de Guzmán, President of
the International Commission on Mathematical Instruction,
in his talk at the 7-th International
Congress on Mathematics Education
[Gu]:

WHAT THE KNOWLEDGE OF THE HISTORY OF MATHEMATICS AND OF THE PARTICULAR SUBJECT
CAN OFFER US:

A human vision of science and of mathematics: not
just truths, methods, techniques
coming from nowhere, not just facts and skills without soul,
without history, but the results
of the efforts of persons motivated by deep interest and passion;
not as godlike science, but
human and so incomplete and fallible; the human side of the great
discoveries and discoverers.

A frame in which all elements appear in their right place:
not facts centuries apart in
their origin presented together in the same bag without a single
remark, but explorations in their own
context and with their own motivation; past fashions in order to
be able to understand present
fashions; the deep connections along time of the different
leitmotivs of the mathematical symphony.

A dynamical vision of the evolution of mathematics:
the motivation and driving forces at
the roots of the ideas and methods of the subject; the
primordial creativity around each
particular subject, its genesis and its progress, . . . .

Jerry Lodder has developed written historical
projects during previous offerings
of an introductory and an intermediate discrete mathematics course.
Students became actively involved in the projects,
writing coherent, mathematically sound papers, and several excelled
beyond the particulars of the given questions. The topics of the
first three projects were chosen from the works of Archimedes
(c. 287--212 B.C.E.) and Blaise Pascal (1623--1662) concerning
the finite sums of powers. For the first project, ``In the Words of
Archimedes,''
students were asked to read Archimedes' verbal statement of the sum of
squares [Di, p. 122]:

If a series of any number of lines be given, which exceed one another
by an equal amount, and the difference be equal to the least, and if
other lines be given equal in number to these, and in quantity to the
greatest, the squares on the lines equal to the greatest, plus the
square on the greatest and the rectangle contained by the least and
the sum of all those exceeding one another by an equal amount will be
the triplicate of all the squares on the lines exceeding one another
by an equal amount.

The project then asked the students to rewrite the statement using
symbolic logic. Such an exercise meshes well with a first chapter on
logic in many discrete mathematics texts, which often include homework
problems such as ``Write the statement `If it is cold and raining,
then I will stay home' in symbols, using the predicates, P: It is
cold, Q: It is raining, R: I will stay home.'' A particular
drawback of modern textbooks is their lack of motivation, perspective
and context. Why not apply symbolic logic to the very statements
which historically drove the development of mathematics, allowing the
students to hone their verbal and deductive skills on significant
problems?

A second project, ``In the Words of Archimedes II,''
built on the first by asking students to verify
Archimedes' statement about the sums of squares using ideas from his
original proof, a proof which did not involve induction. A third
project, ``In the Words of Pascal,''
presented Pascal's bold statement about the sums
of arbitrary powers taken from a translation of Sommation des
Puissances Numériques
[P2],
and asked the students to prove, now using induction, that
the sum of ik from i = 1 to n
is a polynomial in
n of degree k + 1. Although the project did not ask for the
coefficients of this polynomial in explicit form, a few students
became fascinated with this problem, and essentially anticipated the
first three Bernoulli numbers. Homework problems in a standard text
almost never elicit such a response from students. A future project
in this direction will investigate Jakob Bernoulli's (1654--1705)
original work on summations
[Be].

Guram Bezhanishvili has written a major project on Georg Cantor's
(1845--1918) revolutionary ideas concerning infinite sets using
excerpts from Cantor's memoir Contributions to the Founding of
the Theory of Transfinite Numbers
[Cn].
The project, ``Are All Infinities Created Equal?''
(pdf file)
(ps file),
begins by asking the students to identify certain properties of
``naive set theory'' as explicated by Cantor, and then write these
observations using modern notation. The project continues
with a comparison of the cardinality of various infinite sets,
such as the set of rational numbers, and the set of real numbers
in the interval [0, 1], and concludes with the observation that
certain infinite sets are larger than others
(2 to the aleph naught is greater than aleph naught),
a paradigm-breaking idea about the infinite.

Additionally, Jerry has introduced logic in a beginning discrete
mathematics course
[L2]
via Alan Turing's (1912--1954) original paper ``On
Computable Numbers with an Application to the Entscheidungsproblem''
[T].
Through the first project, ``An Introduction
to Turing Machines'' (pdf file)
(ps file), the students witness an
idea which is the forerunner of a programmable computer, a Turing
machine, and in the second, ``Turing Machines, Induction and
Recursion'' (pdf file)
(ps file),
they observe how the concept of recursion
arises naturally when writing a Turing machine to perform a basic operation such
as multiplication of positive integers. Moreover, induction is used
to verify that the output of the machine is correct.
At the conclusion of the Turing
projects, a student from an under represented minority remarked:
``I thought this course would just be
about numbers, but instead, I learned how computers started.''
Another student, when describing the projects, claimed ``That's how I learn.''

The first two Turing projects deal with the construction
and design of Turing machines to perform certain computational tasks,
and remain at an introductory level, primarily
to serve the needs of a beginning course in discrete mathematics.
Of course, Turing's original
motivation was to solve
Hilbert's decision problem,
which Turing masterfully does in
[T],
and in so doing, he develops the idea
of a universal computing machine (known as a compiler or an interpreter
today). The third project in this sequence, ``The Universal Computing
Machine'' (pdf file)
(ps file), outlines Turing's logical
construction of this device, and is well-suited for an intermediate or
an advanced undergraduate course in logic, discrete mathematics, or
computer science. A fourth project in this sequence,
``The Decision Problem'' (pdf file)
(ps file),
offers Turing's (negative) solution to this
issue, which today has become known as the halting problem in computer science.
These last two Turing projects were used recently in an intermediate
discrete mathematics course, at the conclusion of which student
comments reveal: ``I found the two projects to be challenging and
extremely pertinent to computer science.'' ``The two projects that
dealt with the Turing machines and the halting problem gave me insight
and understanding of the origins of computer science.''

During the Autumn semester of 2003 Jerry Lodder developed two projects
for an introductory course in discrete mathematics which trace the
development of binary arithmetic from the Enlightenment to the
electronic age. The first project (pdf file)
(ps file) begins with Gottfried Wilhelm
Leibniz's (1646--1716) work on binary numeration, offering excerpts
form his 1703 publication ``Explication de l'arithmétique
binaire, qui se sert des seuls caractères 0 et 1, avec des
remarques sur son utilité, et sur ce qu'elle donne des
anciennes figures Chinoises de Fohy,''
(An Explanation of Binary Arithmetic Using only the Characters 0 and
1, with Remarks about its Utility and the Meaning it Gives to the
Ancient Chinese Figures of Fuxi)
[Ge, p. 223--227]
[Le].
For Leibniz, binary numbers represented a confluence of several ideas,
including order, harmony, a candidate for his universal
language (lingua generalis), an analogy of creation with 0
denoting nothing and 1 denoting God, and an interpretation of
the ancient Chinese text of divination the Yijing
(I-Ching or Book of Changes) in terms of binary
numeration
[Sw].
Leibniz also cites an ease of calculation with base 2 numbers,
particularly for multiplication and division, which do not require the
memorization of a multiplication table or methods of trial and error,
as is often the case for long division in base 10.

The project continues with a brief account of the Electronic Numerical
Integrator and Computer (ENIAC) developed at the University of
Pennsylvania's Moore School of Electrical Engineering during the years
1943--1945, and discusses its use of base 10 arithmetic, requiring the
storage of a multiplication table for numbers between zero and nine
[Go].
The next generation of computing equipment, the Electronic Discrete
Variable Automatic Computer (EDVAC), heavily influenced by the ideas
of John von Neumann (1903--1957), performed arithmetical calculations
in base 2. The project provides excerpts from von Neumann's 1945
white paper ``First Draft of a Report on the EDVAC''
[vN], in
which he too cites the ease of calculation as an advantage to binary
arithmetic, as well as an economy of representing numerical values
with this system. Moreover, the EDVAC is a model for a ``universal
computing machine'' in the sense of Turing
[T].
This project, ``Binary Arithmetic: From Leibniz to von Neumann''
(pdf file) (ps file)
is ideal for beginning students in discrete mathematics, particularly
those with no previous knowledge of base 2 calculations.

Leibniz's paper on binary arithmetic can be used to motivate and
provide significant material for other topics in discrete mathematics,
such as induction and divisibility properties of integers. Although
these ideas are not developed in the project per se, they could be easily
implemented as classroom activities after completion of the
project. For example,
Leibniz claims that following a study of binary numeration, ``we see the
reason for the famous property of the double geometric progression in
whole numbers, which states that given only one of these numbers in
each degree, we can form all other whole numbers below the double of
the highest degree,'' a statement which today is easily proved by induction.
A present-day formulation of this property might read: Given the values
20, 21, 22, . . . , 2n,
then every positive integer between 1 and 2n+1-1 inclusive
can be (uniquely) expressed as a sum of these powers (with each power
occurring at most once), although a modern formulation of the
statement is best left to the students. Moreover, Leibniz claims that
patterns occur in the binary equivalents of the perfect
squares and other numbers. Asking students to identify a pattern and
then conjecture a corresponding divisibility property of the integers
provides a wonderful discovery exercise.
For example, the second
column in the binary expansion of the perfect squares is comprised
entirely of zeroes, which students are encouraged to reconcile as
4 | n2 (4 divides n2) for
n even, and 4 | (n2-1) for
n odd. Furthermore, examining the zero in the third column
for n2, n odd, leads to the
conjecture that 8 | (n2-1), n odd.

The second project on binary arithmetic ``Arithmetic Backwards from
von Neumann to the Chinese Abacus''
(pdf file) (ps file),
builds on the first, beginning with von Neumann's claim that a
high-speed vacuum tube computer performs arithmetic most efficiently
in base 2. The project then proceeds in a chronologically reverse
order, next examining Claude Shannon's (1916--2001) design of circuits
for computations in Boolean algebra and logic. The project provides
excerpts from Shannon's 1938 paper ``A Symbolic Analysis of Relay and
Switching Circuits''
[Sh],
and asks the students to construct the circuits necessary for binary
addition. A careful study of these examples provides motivation for
writing logic statements in what today is known as ``disjunctive
normal form,'' a concatenation of sub-statements with the connective
``or,'' where each sub-statement contains only negations and the
connective ``and.''

The project continues with the realization that writing a number in
base 2 often requires many digits, and examines an alternative base
that can be easily converted to base 2. Surprisingly a Chinese abacus
provides the ideal tool for computations in base 16, which the
students are asked to realize on their own by exploring all possible
values that can be represented by bead arrangements on a single bar of
this type of abacus. As an in-class activity while the project was
assigned, we discussed base 10 addition and subtraction on an abacus,
with the base 16 analogues of these operations left to the students.
Today hexadecimal notation (base 16) is common place in computer
science, and serves as shorthand for binary numbers, with the
conversion between base 2 and base 16 a simple operation. With
Leibniz's interest in China and the interpretation of the hexagrams of
the Yijing in terms of binary numeration, the Chinese abacus
provides a pleasing confluence of historical, cultural and
computational ideas for the second project, particularly as a tool for
place value arithmetic in a two-power base. Concerning the benefits
of teaching with historical sources, at the conclusion of the course,
students wrote: ``You get an understanding of why you are doing
something.'' ``It ties in better, links can be made.'' ``Gives
meaning to problems.'' ``We can understand why we do certain things
in certain ways.''

David Pengelley has authored a project track to teach induction
from Pascal's ``Treatise on the Arithmetical Triangle,"
(pdf file) (ps file),
where Pascal proves that certain patterns occur in his triangle
via a logical argument that would later become known as induction.
Desh Ranjan has written ``Counting Triangulations of a Polygon,"
(pdf file) (ps file),
which develops the sequence of Catalan numbers from a simple
observation of Lamé. Desh's project contains David's translation
of a letter from Lamé to Liouville, which serves as the primary
source for this work. Hing Leung has completed a project
(pdf file) (ps file)
that outlines the reduction of two-way deterministic finite automata to
one-way automata via the work of Shepherson, and builds on the notion of
a Turing machine. Additionally, Guram Bezhanishvili has outlined
Church's thesis using excerpts from the work of Gödel,
Kleene and Turing, which establish that general recursive functions
coincide with Turing computable functions
(pdf file) (ps file).

The teaching community is invited to use any of the above projects
in relevant mathematics or computer science courses. Typically two or
three projects, chosen around a specific theme, such as
Turing machines, are used in a one-semester course, with the projects
counting for a significant portion (40%--50%) of the course grade. For
each project the students are given one to two weeks, working as an
individual or in groups of two or three, to produce a written paper
addressing the issues raised in the project. Each project usually
replaces an in-class examination. For further general information about the
use of projects see [CG].

♦ ♦
Visit our
new web site for additional projects written as part of the Phase II expansion grant.

ACKNOWLEDGMENT

The development of curricular materials for discrete mathematics
has been partially supported by the National Science Foundation's
Course, Curriculum and Laboratory Improvement Program under grant
DUE-0231113, for which the authors of this web site are most
appreciative. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the authors
and do not necessarily reflect the views of the National Science
Foundation.

[Gr]
Grattan-Guinness, I., The Search for
Mathematical Roots, 1870--1940: Logics, Set Theories and The
Foundations of Mathematics from Cantor through Russell to Gödel,
Princeton University Press, Princeton, 2000.

[L2]
Lodder, J.,
``Introducing Logic via Turing Machines,'' in From
Calculus to Computers: Using the Last 200 Years of Mathematics
History in the Classroom, A. Shell-Gellasch, D. Jardine (eds.),
Mathematical Association of America, Washington, D.C., 2005,
125--133.

[O]
Otero, D., ``An Historical Calculus Course for Liberal
Arts Students'', Newsletter of the International Study Group on the
History and Pedagogy of Mathematics, # 28 (March 1993),
7--9.

[P1]
Pascal, B., ``Treatise on the Arithmetical
Triangle,'' in Great Books of the Western World, M. Adler (ed.),
Encyclopedia Britannica Inc., Chicago, 1991.